# Interaction term

I wanted to ask about interaction term. I am having an ordinal probit model. The two of the independent variables that i have are discrete 1-Uni( a 0,1 dummy variable) and second is continuity (1,2,3) likert scale variable. Is an interaction term of these two variables possible> I hope i have made it clear.

Another problem remains that if i introduce this interaction term (uni*continuity) it remains insignificant, and also makes the variable "uni" insignificant,

I have encountered this problem recently and have not found many sources which explain this so I thought I would answer. The interaction effect in a non-linear model is possible to compute but it is tricky as it is not equal to the marginal effect on the interaction term.

Say you have the probit model $E(y|x_1,x_2)=\Phi(\beta_1 x_1 + \beta_2 x_2 + \beta_{12} x_1 \times x_2)$ where $x_1$ is continuous and $x_2$ is discrete. Interpreting the marginal effect of the interaction term $\beta_{12} \Phi'(.)$ as the true interaction effect is incorrect (this would be true if we had a linear model).

If you take the second cross-partial derivative $\frac{\delta E^2(y|x_1,x_2)}{\delta x_1 \Delta x_2}$ you can find the interaction effect which is

$$=\dfrac{\delta E^2(y|x_1, x_2)}{\delta x_1 \Delta x_2} = \Phi ' (\beta_1 x_1 + \beta_2 + \beta_{12} x_1)(\beta_1 + \beta_{12}) - \Phi ' (\beta_1 x_1)(\beta_1)$$

which as you can see is different from $\beta_{12} \Phi'(.)$. Note that even in cases where your coefficient interaction term is 0, the true interaction effect might still be non zero. Ai & Norton (2002). The other problem with this non-linear interaction effect is that a standard t-test on the coefficient estimate doesn't work and other more advanced methods of calculating the standard errors must be used (see the paper for how they do it).

Ai & Norton (2003) have created a user-written STATA command inteff to visually plot the interaction effect but this is only available after Probit & Logit models unfortunately.

You could, if you are using STATA, calculate the marginal effects using the command margins, dydx(x1) ... at (x2 = 0(1)1) and this will give you the marginal effect when x2=1 and x2=0 which you can compare by supplementing pwcompare(effects) at the end of your code. This however would be the interaction effect for a specific value of your $x$s (The real interaction effect varies by individual as their $\Phi(.)$ changes).

Another way to represent the interaction effect is graphically and this is explained by Greene (2010).